1) Evaluate the double integral ∬Dx^2 y dA, where D is the top half of the...

1) Evaluate the double integral ∬Dx^2 y dA, where D is the top half of the disc with center the origin and radius 2 by changing to polar coordinates.

2) Use a double integral to find the area of one loop of the rose r= 6cos (3θ).

3) Use polar coordinates to find the volume of the solid below the paraboloid z=50−2x^2−2y^2 and above the xy-plane.

Expert Solution

For the given region D, we have, In polar coordinates, x=2 cos and y= 2 sin

Where varies from 0 to and r varies from 0 to 2. The step by step explanatory solution is provided below.

milcah answered 2 years ago

using / for integral Evaluate the double integral //R cos( (y-x)/(y+x) )dA where R is the...

using / for integral Evaluate the double integral //R cos( (y-x)/(y+x) )dA where R is the trapezoidal region with vertices (1,0), (2,0), (0,2), and (0,1)

evaluate the integral by making an appropriate change of variables double integral of 5sin(25x^2+64y^2) dA, where...

evaluate the integral by making an appropriate change of variables double integral of 5sin(25x^2+64y^2) dA, where R is the region in the first quadrant bounded by the ellipse 25x^2 +64y^2=1

Calculate the double integral ∫∫Rxcos(x+y)dA∫∫Rxcos⁡(x+y)dA where RR is the region: 0≤x≤π3,0≤y≤π2

Find the double integral of region (2x-y)dA, where region R is in the first quadrant enclosed...

Find the double integral of region (2x-y)dA, where region R is in the first quadrant enclosed by the circle x2+y2=4 and the lines x=0 and y=x

1.) Evaluate the given definite integral. Integral from 4 to 5 dA∫45 (0.2e^−0.2A +3/A) dA 2.)...

1.) Evaluate the given definite integral. Integral from 4 to 5 dA∫45 (0.2e^−0.2A +3/A) dA 2.) Evaluate the definite integral. Integral from negative 1 to 1 dx∫−1 1 (x^2+1) dx 3.) Evaluate the definite integral. Integral from 0 to 2 dx∫02 (2x^2+x+6) dx 4.) Evaluate the definite integral. Integral from 1 to 4 left dx∫14 (x^3/2+x^1/2−x^−1/2) dx 5.) Evaluate the definite integral. Integral from negative 2 to negative 1 dx∫−2−1 (3x^−4) dx

Evaluate the integral. ∫4−2. (1/(x+5)((x^2)+16)) dx

Use the given transformation to evaluate the integral. (12x + 8y) dA R , where R...

Use the given transformation to evaluate the integral. (12x + 8y) dA R , where R is the parallelogram with vertices (−1, 3), (1, −3), (2, −2), and (0, 4) ; x = 1/ 4 (u + v), y = 1/ 4 (v − 3u)

Evaluate the integral. (Use C for the constant of integration.) (x^2-1)/(sqrt(25+x^2)*dx Evaluate the integral. (Use C...

Evaluate the integral. (Use C for the constant of integration.) (x^2-1)/(sqrt(25+x^2)*dx Evaluate the integral. (Use C for the constant of integration.) dx/sqrt(9x^2-16)^3 Evaluate the integral. (Use C for the constant of integration.) 3/(x(x+2)(3x-1))*dx

1). Find the dervatives dy/dx and d^2/dx^2 , and evaluate them at t = 2. x...

1). Find the dervatives dy/dx and d^2/dx^2 , and evaluate them at t = 2. x = t^2 , y= t ln t 2) Find the arc length of the curve on the given interval. x = ln t , y = t + 1 , 1 < or equal to t < or equal to 2 3) Find the area of region bounded by the polar curve on the given interval. r = tan theta , pi/6 < or...

Evaluate the integral: ∫−14 / x^2√x^2+100 dx (A) Which trig substitution is correct for this integral?...

Evaluate the integral: ∫−14 / x^2√x^2+100 dx (A) Which trig substitution is correct for this integral? x=−14sec(θ) x=100sec(θ) x=10tan(θ) x=100sin(θ) x=10sin(θ) x=100tan(θ) (B) Which integral do you obtain after substituting for x and simplifying? Note: to enter θ, type the word theta. (C) What is the value of the above integral in terms of θ? (D) What is the value of the original integral in terms of x?

Question